## Annals of Probability

### Kolmogorov's test for super-Brownian motion

#### Abstract

We prove a Kolmogorov test for super-Brownian motion started at the Dirac mass at the origin. More precisely, we determine the functions $g$ such that for all $t$ small enough, the support of the process at time $t$ will be contained in the ball of radius $g(t)$ centered at 0. As a consequence, we get a necessary and sufficient condition for the existence in certain space-time domains of a solution of the associated semilinear partial differential equation that blows up at the origin.

#### Article information

Source
Ann. Probab., Volume 26, Number 3 (1998), 1041-1056.

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022855744

Digital Object Identifier
doi:10.1214/aop/1022855744

Mathematical Reviews number (MathSciNet)
MR1634414

Zentralblatt MATH identifier
0938.60087

#### Citation

Dhersin, Jean-Stéphane; Le Gall, Jean-François. Kolmogorov's test for super-Brownian motion. Ann. Probab. 26 (1998), no. 3, 1041--1056. doi:10.1214/aop/1022855744. https://projecteuclid.org/euclid.aop/1022855744

#### References

• [1] Chung, K. L., Erd ¨os, P. and Sirao, T. (1959). On the Lipschitz condition for Brownian motion. J. Math. Soc. Japan 11 263-274.
• [2] Dawson, D. A., Hochberg, K. J. and Vinogradov, V. (1996). High-density limits of hierarchically structured branching-diffusion populations. Stochastic Process. Appl. 62 191-222.
• [3] Dawson, D. A., Iscoe, I. and Perkins, E. A. (1989). Super-Brownian motion: path properties and hitting probabilities. Probab. Theory Related Fields 83, 135-205.
• [4] Dawson, D. A. and Vinogradov, V. (1994). Almost sure path properties of 2 d -superprocesses. Stochastic Process. Appl. 51 221-258.
• [5] Dhersin, J. S. (1997). Super-mouvement brownien, serpent brownien et ´equations aux d´eriv´ees partielles. Th ese, Univ. Paris VI.
• [6] Dhersin, J. S. (1998). Lower functions for the support of super-Brownian motion. Preprint.
• [7] Dhersin, J. S. and Le Gall, J. F. (1997). Wiener's test for super-Brownian motion and the Brownian snake. Probab. Theory Related Fields 108 103-129.
• [8] Dynkin, E. B. (1991). Branching particle systems and superprocesses. Ann. Probab. 19 1157-1194.
• [9] Dynkin, E. B. (1992). Superdiffusions and parabolic nonlinear differential equations. Ann. Probab. 20 942-962.
• [10] Dynkin, E. B. (1993). Superprocesses and partial differential equations. Ann. Probab. 21 1185-1262.
• [11] It o, K. and McKean, H. P. (1965). Diffusion Processes and Their Sample Paths. Springer, Berlin.
• [12] Le Gall, J. F. (1993). A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 25-46.
• [13] Le Gall, J. F. (1994). A path-valued Markov process and its connections with partial differential equations. In: Proceedings of the First European Congress of Mathematics (A. Joseph, F. Mignot, F. Murat, B. Prum and R. Rentschler, eds.) 2 185-212. Birkh¨auser, Boston.
• [14] Le Gall, J. F. (1995). The Brownian snake and solutions of u = u2 in a domain. Probab. Theory Related Fields 102 393-432.
• [15] Le Gall, J. F. and Perkins, E. A. (1995). The Hausdorff measure of the support of twodimensional super-Brownian motion. Ann. Probab. 23 1719-1747.
• [16] Tribe, R. (1989) Path properties of superprocesses. Ph.D. dissertation, Univ. British Columbia.